Page 382 - Introduction to Continuum Mechanics

366 Streamline, Pathline, Streakline, Steady, Unsteady, Laminar and Turbulent Flow

6.10 Streamline, Pathline, Streakline, Steady, Unsteady, Laminar and
Turbulent Flow
(a) Streamline.
A streamline at time Ms a curve whose tangent at every point has the direction of the
instantaneous velocity vector of the particle at the point. Experimentally, streamlines on the
surface of a fluid are often obtained by sprinkling it with reflecting particles and making a
short-time exposure photograph of the surface. Each reflecting particle produces a short line
on the photograph approximating the tangent to a streamline. Mathematically, streamlines
can be obtained from the velocity field v(x,f) as follows:
Let x = x(s) be the parametric equation for the streamline at time t, which passes through
a given point X0. Then an s can always be chosen such that

Given the velocity field in dimensionless form

find the streamline which passes through the point (a^o^s) at time t

Solution. From

we have

Thus,

i.e.,

t The example is chosen to demonstrate the differences between streamlines, pathlines and streaktines. The
velocity field obviously does not correspond to an incompressible fluid.